1. About the project

Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you can use R Markdown syntax.

The first comment

I’m feeling excited to start this new course. I am expecting to learn how to use R and apply it for my studies in social psychology. I think with the current amount of data that is collected everyday, social sciences as a whole has great chances of improving. I first knew about this course after contacting Kimmo, as recommended by the coordinator of my master’s programme.

2. Regression and Model Validation

Describe the work you have done this week and summarize your learning.

Reading the data

lrn14 <- read.table("data/learning2014.txt", header = TRUE, sep = "\t")

str(lrn14)  
## 'data.frame':    166 obs. of  7 variables:
##  $ gender  : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
##  $ age     : int  53 55 49 53 49 38 50 37 37 42 ...
##  $ attitude: int  37 31 25 35 37 38 35 29 38 21 ...
##  $ deep    : num  3.58 2.92 3.5 3.5 3.67 ...
##  $ stra    : num  3.38 2.75 3.62 3.12 3.62 ...
##  $ surf    : num  2.58 3.17 2.25 2.25 2.83 ...
##  $ points  : int  25 12 24 10 22 21 21 31 24 26 ...
dim(lrn14)
## [1] 166   7

As we can see, we have here a data frame imported as lrn14. The data reflects the relationship between learning approaches and the students achievements. This dataset includes 166 observations with 7 variables each. The variables are gender, age, attitude, average of the deep approach, average of the strategic approach, average of the surface approachs and the points obtained in the exam.

Exploring the data

pairs(lrn14[-1])

summary(lrn14)
##  gender       age           attitude          deep            stra      
##  F:110   Min.   :17.00   Min.   :14.00   Min.   :1.583   Min.   :1.250  
##  M: 56   1st Qu.:21.00   1st Qu.:26.00   1st Qu.:3.333   1st Qu.:2.625  
##          Median :22.00   Median :32.00   Median :3.667   Median :3.188  
##          Mean   :25.51   Mean   :31.43   Mean   :3.680   Mean   :3.121  
##          3rd Qu.:27.00   3rd Qu.:37.00   3rd Qu.:4.083   3rd Qu.:3.625  
##          Max.   :55.00   Max.   :50.00   Max.   :4.917   Max.   :5.000  
##       surf           points     
##  Min.   :1.583   Min.   : 7.00  
##  1st Qu.:2.417   1st Qu.:19.00  
##  Median :2.833   Median :23.00  
##  Mean   :2.787   Mean   :22.72  
##  3rd Qu.:3.167   3rd Qu.:27.75  
##  Max.   :4.333   Max.   :33.00

In the first representation we used the pairs() function to show the distribution of the different variables as a function of the others.

reg_model <- lm(points ~ attitude + stra + surf, data = lrn14)
summary(reg_model)
## 
## Call:
## lm(formula = points ~ attitude + stra + surf, data = lrn14)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.1550  -3.4346   0.5156   3.6401  10.8952 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.01711    3.68375   2.991  0.00322 ** 
## attitude     0.33952    0.05741   5.913 1.93e-08 ***
## stra         0.85313    0.54159   1.575  0.11716    
## surf        -0.58607    0.80138  -0.731  0.46563    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.296 on 162 degrees of freedom
## Multiple R-squared:  0.2074, Adjusted R-squared:  0.1927 
## F-statistic: 14.13 on 3 and 162 DF,  p-value: 3.156e-08

We can observe that in this multiple regression analysis we obtain a p-value that is very close to 0. This means that in this model, our explanatory variables are very likely to explain the dependent variable.
The R-Squared represents the proportion of the variance of the points that is explained by the explanatory variables. In this case, our model explains approximately a 20% of the variance of the points.

par(mfrow = c(2,2))
plot(reg_model, which = c(1,2,5))

We can observe from some of the plots that the model represents a good fit. As an example, in the QQ-Plot, the great majority of the values fit very close to the line.
On the other hand, in the plot between the Residuals and the Fitted values, we can’t observe a major disperstion of the results as the fitted values increased.

3. Logistic regression

For this exercise on logistic regression, we will be will be working on data from two questionnaires related to student performance. This data approaches student achievement in secondary education of two Portuguese schools. The data attributes include student grades, demographic, social and school related features) and it was collected by using school reports and questionnaires From this questionnaires, we are particularly interested in the data related to alcohol consumption.

Reading the data

We will use a prepared data extracted from the previously mentioned data set. We modified it to obtain information relevant for our analysis on alcohol consumption. We will start reading the table, storing it into the variable “alc” and showing the names of the different columns.

alc <- read.table("data/alc.csv", header = TRUE, sep = "\t")
colnames(alc)
##  [1] "school"     "sex"        "age"        "address"    "famsize"   
##  [6] "Pstatus"    "Medu"       "Fedu"       "Mjob"       "Fjob"      
## [11] "reason"     "nursery"    "internet"   "guardian"   "traveltime"
## [16] "studytime"  "failures"   "schoolsup"  "famsup"     "paid"      
## [21] "activities" "higher"     "romantic"   "famrel"     "freetime"  
## [26] "goout"      "Dalc"       "Walc"       "health"     "absences"  
## [31] "G1"         "G2"         "G3"         "alc_use"    "high_use"

Hypothesis

For our analysis we will choose the variables sex, Medu, Fedu and activities. We picked the mother’s education and the father’s education (Medu and Fedu) as education has been shown to play an important role in very different aspects of life and health. Finally, we will pick the absences variable as we assume it could be linked to high alcohol consumption.

Exploring the distributions

library(tidyr)
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(ggplot2)
library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
## 
## Attaching package: 'GGally'
## The following object is masked from 'package:dplyr':
## 
##     nasa
#We isolate the selected variables according to our hypothesis
pick <- c("high_use","sex","Medu","Fedu","absences")
alc_pick <- select(alc, one_of(pick))

#We create some plots with the variables
str(alc_pick)
## 'data.frame':    382 obs. of  5 variables:
##  $ high_use: logi  FALSE FALSE TRUE FALSE FALSE FALSE ...
##  $ sex     : Factor w/ 2 levels "F","M": 1 1 1 1 1 2 2 1 2 2 ...
##  $ Medu    : int  4 1 1 4 3 4 2 4 3 3 ...
##  $ Fedu    : int  4 1 1 2 3 3 2 4 2 4 ...
##  $ absences: int  5 3 8 1 2 8 0 4 0 0 ...
gather(alc_pick) %>% glimpse()
## Warning: attributes are not identical across measure variables;
## they will be dropped
## Observations: 1,910
## Variables: 2
## $ key   <chr> "high_use", "high_use", "high_use", "high_use", "high_use", "hi…
## $ value <chr> "FALSE", "FALSE", "TRUE", "FALSE", "FALSE", "FALSE", "FALSE", "…
gather(alc_pick) %>% ggplot(aes(value)) + facet_wrap("key", scales="free") + geom_bar()
## Warning: attributes are not identical across measure variables;
## they will be dropped

p1 <- ggplot(alc_pick, aes(sex, fill=high_use)) 
p2 <- ggplot(alc_pick, aes(Medu, fill=high_use))
p3 <- ggplot(alc_pick, aes(Fedu, fill=high_use))
p4 <- ggplot(alc_pick, aes(x=high_use, y=absences))

p1 + geom_bar()

p2 + geom_bar()

p3 + geom_bar()

p4 + geom_boxplot()

table(alc_pick$high_use, alc_pick$sex)
##        
##           F   M
##   FALSE 156 112
##   TRUE   42  72
table(alc_pick$high_use, alc_pick$Medu)
##        
##          0  1  2  3  4
##   FALSE  1 33 80 59 95
##   TRUE   2 18 18 36 40
table(alc_pick$high_use, alc_pick$Fedu)
##        
##          0  1  2  3  4
##   FALSE  2 53 75 72 66
##   TRUE   0 24 30 27 33
table(alc_pick$high_use, alc_pick$absences)
##        
##          0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 16 17 18 19 20 21 26 27 29
##   FALSE 52 38 42 33 24 16 16  9 14  6  5  2  4  1  1  0  0  1  0  2  1  0  0  0
##   TRUE  13 13 16  8 12  6  5  3  6  6  2  4  4  1  6  1  1  1  1  0  1  1  1  1
##        
##         44 45
##   FALSE  0  1
##   TRUE   1  0

For what we can see in this initial explorations, the sex variable seem to have a some relevance. Fedu (father’s education) shows similar results in both groups. However, Medu (mother’s education) and absences seem to have some indications of influence on alcohol consumption on young people.

Logistic regression

m <- glm(data=alc_pick, high_use ~ sex + Medu + Fedu + absences, family="binomial")
summary(m)
## 
## Call:
## glm(formula = high_use ~ sex + Medu + Fedu + absences, family = "binomial", 
##     data = alc_pick)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.2692  -0.8616  -0.6186   1.0873   2.0814  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.72690    0.37465  -4.609 4.04e-06 ***
## sexM         1.00307    0.24239   4.138 3.50e-05 ***
## Medu        -0.13932    0.14599  -0.954    0.340    
## Fedu         0.10037    0.14230   0.705    0.481    
## absences     0.09866    0.02327   4.240 2.23e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 465.68  on 381  degrees of freedom
## Residual deviance: 429.14  on 377  degrees of freedom
## AIC: 439.14
## 
## Number of Fisher Scoring iterations: 4

The results show that just sex is a relevant variable from the ones we chose. We will build a new model with just the sex variable.

m2 <- glm(data=alc_pick, high_use ~ sex + absences, family="binomial")
summary(m2)
## 
## Call:
## glm(formula = high_use ~ sex + absences, family = "binomial", 
##     data = alc_pick)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.2753  -0.8753  -0.6081   1.0921   1.9920  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -1.83606    0.22251  -8.252  < 2e-16 ***
## sexM         0.97762    0.23982   4.076 4.57e-05 ***
## absences     0.09659    0.02306   4.189 2.80e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 465.68  on 381  degrees of freedom
## Residual deviance: 430.07  on 379  degrees of freedom
## AIC: 436.07
## 
## Number of Fisher Scoring iterations: 4
OR <- coef(m2) %>% exp
CI <- confint(m2) %>% exp
## Waiting for profiling to be done...
cbind(OR, CI)
##                   OR     2.5 %    97.5 %
## (Intercept) 0.159445 0.1012577 0.2427684
## sexM        2.658116 1.6710354 4.2863129
## absences    1.101409 1.0549317 1.1548057

According to these results, males are almost 2.6 times more likely to have a high alcohol consumption than females. Absences are also linked. the data showing that high alcohol consumption probabilities grows as absences increase.

prob <- predict(m2, type="response")
alc_pick <- mutate (alc_pick, probs = prob)
alc_pick <- mutate(alc_pick, prediction = probs > .5)
table(high_use = alc_pick$high_use, prediction = alc_pick$prediction)
##         prediction
## high_use FALSE TRUE
##    FALSE   258   10
##    TRUE     88   26
# plot
ggplot(alc_pick, aes(x = probs, y = high_use, col = prediction)) + geom_point()

4. Clustering and classification

First look at the data

data("Boston")
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim(Boston)
## [1] 506  14

The Boston dataset contains 506 rows and 14 columns of information relevant to the housing values in the suburbs of Boston.

Graphical analysis

#We show a summary of the dataset
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
#We create a correlation matrix and store it
cor_matrix<-cor(Boston)
cor_matrix %>% round(digits = 2)
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax ptratio
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58    0.29
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31   -0.39
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72    0.38
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04   -0.12
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67    0.19
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29   -0.36
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51    0.26
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53   -0.23
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91    0.46
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00    0.46
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46    1.00
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44   -0.18
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54    0.37
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47   -0.51
##         black lstat  medv
## crim    -0.39  0.46 -0.39
## zn       0.18 -0.41  0.36
## indus   -0.36  0.60 -0.48
## chas     0.05 -0.05  0.18
## nox     -0.38  0.59 -0.43
## rm       0.13 -0.61  0.70
## age     -0.27  0.60 -0.38
## dis      0.29 -0.50  0.25
## rad     -0.44  0.49 -0.38
## tax     -0.44  0.54 -0.47
## ptratio -0.18  0.37 -0.51
## black    1.00 -0.37  0.33
## lstat   -0.37  1.00 -0.74
## medv     0.33 -0.74  1.00
#We create a correlation plot
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6)

#We create a distribution plot
Boston %>% gather() %>% ggplot(aes(value)) + facet_wrap(~ key, scales = "free") + geom_density(colour="red")

We can easily observe thanks to the correlation plot the different interactions between the elements of the dataset. We also see in distribution graphics how the different variables do not represent a normal distribution.

Standarization and creation of the categorical variable

#We center and standarized the variables
boston_scaled <- scale(Boston)
#We show a summary of the scaled variables
summary(boston_scaled)
##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865
# We show the class of the boston_scaled object
class(boston_scaled)
## [1] "matrix"
#We change the object to data frame to work later with it
boston_scaled <- as.data.frame(boston_scaled)
bins <- quantile(boston_scaled$crim)
#We create a categorical variable of the crime rate in the Boston dataset (from the scaled crime rate).
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label = c("low","med_low","med_high","high"))
#We remove the old crime rate variable from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)
#We divide the dataset to train and test sets, so that 80% of the data belongs to the train set. 
n <- nrow(boston_scaled)
#We choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)
#We create train set
train <- boston_scaled[ind,]
#We create test set 
test <- boston_scaled[-ind,]

Fitting the linear discriminant analsysis on the train set

#We use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables
lda.fit <- lda(crime ~ ., data = train)

# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(train$crime)

# We draw the plot with the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)

Prediction of the classes

#We save the correct classes from test data
correct_classes <- test$crime
#We remove the crime variable from test data
test <- dplyr::select(test, -crime)
#We predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
#We cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       14      13        1    0
##   med_low    2      13        8    0
##   med_high   1       4       19    1
##   high       0       0        0   26

As we can observe from the cross tabulation, our model predicts best the category of high. Second, the category of low is also high enough. Finally, the med_low and med_high categories have lower accuracy of predictions, even though the majority is still correct.

Distances and k-means

#We load the data and standarize it
data(Boston)
boston_scaled <- scale(Boston)
#We calculate the distances with the Eucidean distance
dist(boston_scaled)
#We run k-means algorithm on the dataset.
km <-kmeans(boston_scaled, centers = 2)
#We visualize the clusters
pairs(boston_scaled, col = km$cluster)

From the options we can choose, 2 centers seem to be the best representation of the results, as we can see clearly differentiated clusters with such number. With more centers, the clusters are not separated properly and they tend to occupy the same space of our plots.

Super-bonus

library(plotly)
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
model_predictors <- dplyr::select(train, -crime)

# check the dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)

plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = classes)

5. Dimensionality reduction techniques

Graphical overview and summaries

library(corrplot)
library(dplyr)
library(ggplot2)
library(tidyr)
library(GGally)
human <- read.csv(file ="~/IODS-project/data/human.csv", row.names = 1)
summary(human)
##      edu2FM           labFM           lifeExp       educationExp  
##  Min.   :0.1717   Min.   :0.1857   Min.   :49.00   Min.   : 5.40  
##  1st Qu.:0.7264   1st Qu.:0.5984   1st Qu.:66.30   1st Qu.:11.25  
##  Median :0.9375   Median :0.7535   Median :74.20   Median :13.50  
##  Mean   :0.8529   Mean   :0.7074   Mean   :71.65   Mean   :13.18  
##  3rd Qu.:0.9968   3rd Qu.:0.8535   3rd Qu.:77.25   3rd Qu.:15.20  
##  Max.   :1.4967   Max.   :1.0380   Max.   :83.50   Max.   :20.20  
##       gni             matMor           birth           repParl     
##  Min.   :   581   Min.   :   1.0   Min.   :  0.60   Min.   : 0.00  
##  1st Qu.:  4198   1st Qu.:  11.5   1st Qu.: 12.65   1st Qu.:12.40  
##  Median : 12040   Median :  49.0   Median : 33.60   Median :19.30  
##  Mean   : 17628   Mean   : 149.1   Mean   : 47.16   Mean   :20.91  
##  3rd Qu.: 24512   3rd Qu.: 190.0   3rd Qu.: 71.95   3rd Qu.:27.95  
##  Max.   :123124   Max.   :1100.0   Max.   :204.80   Max.   :57.50
ggplot(gather(human), aes(value)) + facet_wrap(~ key, scales = "free") + geom_density(fill="#FF9999", colour="black")

ggpairs(human)

cor(human) %>% corrplot(type = "upper")

We have eight numeric variables. In the graphical representations, we can see that in general the different values show normal distributions, accumulating the majority of the cases around a central point. We observe how there are some negative and positive relations between different values, like adolescent birth rate and maternal mortality ratio.

Performing Principal Component Analysis (PCA)

PCA on the not standardized human data

pca_human <- prcomp(human)
biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col = c("grey40", "deeppink2"))
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

Because the PCA is supposed to be done with standardized values, we get some warnings. The results we obtain are related to the scale of the values. Higher scales (like in the case of the GNI index that we modified) results in higher relevance in the plot.

Performing Principal Component Analysis (PCA) with standardized values

# We standardized the variables
human_std <- scale(human)
pca_humanstd <- prcomp(human_std)
s <- summary(pca_humanstd)
s
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5     PC6     PC7
## Standard deviation     2.0708 1.1397 0.87505 0.77886 0.66196 0.53631 0.45900
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595 0.02634
## Cumulative Proportion  0.5361 0.6984 0.79413 0.86996 0.92473 0.96069 0.98702
##                            PC8
## Standard deviation     0.32224
## Proportion of Variance 0.01298
## Cumulative Proportion  1.00000
pca_pr <- round(100*s$importance[2, ], digits = 1)
pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)")
biplot(pca_humanstd, cex = c(0.8, 1), col = c("grey40", "red"), xlab = pc_lab[1], ylab = pc_lab[2])

As we mentioned, it was impossible to observe useful information from the biplot with the non-standardized values. However, with this new biplot, we are able to see clearly some information. We observe, for example, how the first Principal Component, situated in the X axis, correposponds to the 53.6% of the total variance, while the second Principal Component, situated in the Y axis, correspondes to the 16,2% of the total variance in the original variables.
We can also observe how the original variables are easily distributed in two directions (almost in parallel to the axis). For example, repPar and labFM are in the same direction than the PC2, while the other variables are in the direction of the PC1. We also see how the angles between the variables have for the most part small angles between each, which demonstrates a high correlation (this correlation can be positive or negative).
Being more specific, we can observe how the maternal mortality is highly correlated with the adolescent birth rate. We also observe how the ratio of female workers over men is linked with the number of female parliamentary representatives. And we can conclude also that these two variables have no correlation with the maternal mortality or the adolescent birth rate. Finally, we can observe how the life expectancy, the expected education levels, the GNI ratio or the ratio of female over male education have a high degree of positive correlation with each other and have a strong negative correlation with the adolescent birth rate and maternal mortality rate.

Interpretation of components

The first Principal Component seems to represent a value of the general health and life conditions and gender equality in the domestic sphere.
On the other hand, the second Principal Component seems to represent the integration of women in the public sphere (politics and jobs).

Multiple Correspondence Analysis

library(FactoMineR)
data(tea)
str(tea)
## 'data.frame':    300 obs. of  36 variables:
##  $ breakfast       : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
##  $ tea.time        : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
##  $ evening         : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
##  $ lunch           : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dinner          : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
##  $ always          : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
##  $ home            : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
##  $ work            : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
##  $ tearoom         : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ friends         : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
##  $ resto           : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
##  $ pub             : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Tea             : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How             : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ sugar           : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ how             : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ where           : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ price           : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
##  $ age             : int  39 45 47 23 48 21 37 36 40 37 ...
##  $ sex             : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ SPC             : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
##  $ Sport           : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
##  $ age_Q           : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
##  $ frequency       : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
##  $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
##  $ spirituality    : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
##  $ healthy         : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
##  $ diuretic        : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
##  $ friendliness    : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
##  $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
##  $ feminine        : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
##  $ sophisticated   : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
##  $ slimming        : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ exciting        : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
##  $ relaxing        : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
##  $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
dim(tea)
## [1] 300  36
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch","pub","always")
tea_time <- select(tea, one_of(keep_columns))
str(tea_time)
## 'data.frame':    300 obs. of  8 variables:
##  $ Tea   : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How   : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ how   : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ pub   : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ always: Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped

mca <- MCA(tea_time, graph = TRUE)

summary(mca)
## 
## Call:
## MCA(X = tea_time, graph = TRUE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6   Dim.7
## Variance               0.212   0.204   0.173   0.147   0.141   0.127   0.120
## % of var.             13.076  12.545  10.654   9.025   8.682   7.814   7.371
## Cumulative % of var.  13.076  25.621  36.275  45.300  53.982  61.796  69.167
##                        Dim.8   Dim.9  Dim.10  Dim.11  Dim.12  Dim.13
## Variance               0.106   0.106   0.093   0.086   0.063   0.046
## % of var.              6.546   6.518   5.737   5.293   3.880   2.858
## Cumulative % of var.  75.714  82.232  87.969  93.262  97.142 100.000
## 
## Individuals (the 10 first)
##                       Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1                  | -0.446  0.311  0.228 |  0.229  0.086  0.060 |  0.250
## 2                  | -0.406  0.259  0.131 |  0.202  0.067  0.033 |  0.656
## 3                  | -0.457  0.327  0.387 |  0.143  0.033  0.038 |  0.110
## 4                  | -0.547  0.469  0.538 |  0.044  0.003  0.004 | -0.210
## 5                  | -0.344  0.186  0.166 |  0.049  0.004  0.003 | -0.119
## 6                  | -0.457  0.327  0.387 |  0.143  0.033  0.038 |  0.110
## 7                  | -0.457  0.327  0.387 |  0.143  0.033  0.038 |  0.110
## 8                  | -0.406  0.259  0.131 |  0.202  0.067  0.033 |  0.656
## 9                  |  0.287  0.129  0.058 | -0.451  0.333  0.145 |  0.461
## 10                 |  0.439  0.303  0.147 | -0.141  0.032  0.015 |  0.835
##                       ctr   cos2  
## 1                   0.120  0.072 |
## 2                   0.829  0.343 |
## 3                   0.023  0.022 |
## 4                   0.085  0.079 |
## 5                   0.027  0.020 |
## 6                   0.023  0.022 |
## 7                   0.023  0.022 |
## 8                   0.829  0.343 |
## 9                   0.410  0.151 |
## 10                  1.342  0.530 |
## 
## Categories (the 10 first)
##                        Dim.1     ctr    cos2  v.test     Dim.2     ctr    cos2
## black              |   0.279   1.132   0.026   2.764 |   0.339   1.737   0.038
## Earl Grey          |  -0.095   0.340   0.016  -2.202 |  -0.329   4.275   0.195
## green              |  -0.072   0.033   0.001  -0.436 |   1.165   9.162   0.168
## alone              |  -0.105   0.420   0.020  -2.470 |   0.197   1.545   0.072
## lemon              |   0.956   5.914   0.113   5.812 |  -0.343   0.793   0.015
## milk               |  -0.293   1.058   0.023  -2.609 |  -0.257   0.852   0.018
## other              |   0.814   1.169   0.020   2.475 |  -1.209   2.687   0.045
## tea bag            |  -0.675  15.187   0.596 -13.346 |   0.015   0.008   0.000
## tea bag+unpackaged |   0.690   8.785   0.217   8.064 |  -0.673   8.692   0.206
## unpackaged         |   1.385  13.536   0.261   8.842 |   1.683  20.846   0.386
##                     v.test     Dim.3     ctr    cos2  v.test  
## black                3.353 |   1.098  21.454   0.394  10.860 |
## Earl Grey           -7.645 |  -0.433   8.714   0.338 -10.059 |
## green                7.085 |   0.072   0.041   0.001   0.437 |
## alone                4.640 |  -0.042   0.083   0.003  -0.988 |
## lemon               -2.084 |  -0.882   6.172   0.096  -5.359 |
## milk                -2.293 |   0.245   0.910   0.016   2.184 |
## other               -3.675 |   2.426  12.753   0.182   7.379 |
## tea bag              0.306 |  -0.106   0.459   0.015  -2.095 |
## tea bag+unpackaged  -7.856 |   0.350   2.772   0.056   4.089 |
## unpackaged          10.748 |  -0.414   1.482   0.023  -2.641 |
## 
## Categorical variables (eta2)
##                      Dim.1 Dim.2 Dim.3  
## Tea                | 0.026 0.247 0.418 |
## How                | 0.146 0.096 0.276 |
## how                | 0.638 0.482 0.065 |
## sugar              | 0.028 0.032 0.283 |
## where              | 0.714 0.586 0.102 |
## lunch              | 0.010 0.045 0.013 |
## pub                | 0.100 0.117 0.097 |
## always             | 0.039 0.026 0.131 |
plot.MCA(mca, invisible=c("var","quali.sup"), cex=0.7)

plot.MCA(mca, invisible=c("ind"), cex=0.7)

plot(mca, invisible=c("quali.sup"), habillage= "quali")

We can observe in the different plots the variables that define the created dimensions. We also observe the relation between the different variables. The variables that are closer in the space in the plot representation have higher correlation. For instance, in the first MCA factor map, we see how the values tea shop and unpackaged are closer to each other than to any other value. This explaine the connection that they have.